ORF 544 Stochastic Optimization and Learning · » Parametric models • Linear in the parameters...

ORF 544

Stochastic Optimization and LearningSpring, 2019

Warren PowellPrinceton University

http://www.castlelab.princeton.edu

Week 2

Chapter 3: Recursive learning

Overview

Types of learning problemsTypes of approximation strategies

Learning problems

Classes of learning problems in stochastic optimization1) Approximating the objective

.2) Designing a policy .3) A value function approximation

.4) Designing a cost function approximation:

• The objective function 𝐶̅ 𝑆 , 𝑥 |𝜃 .• The constraints 𝑋 𝑆 |𝜃

5) Approximating the transition function

Approximation strategies

Approximation strategies» Lookup tables

• Independent beliefs • Correlated beliefs

» Linear parametric models• Linear models • Sparse-linear• Tree regression

» Nonlinear parametric models• Logistic regression• Neural networks

» Nonparametric models• Gaussian process regression• Kernel regression• Support vector machines• Deep neural networks

Approximating strategies

Lookup tables

» Independent beliefs

» Correlated beliefs• A few dozen observations can

teach us about thousands of points.

» Hierarchical models• Create beliefs at different levels of

aggregation and then use weighted combinations

1( , ) ,...,nx MF x W x x x

Parametric models» Linear models

• Might include sparse-additive, where many parameters are zero.

» Nonlinear models

• (Shallow) Neural networks

| ( )f ff F

( ) ...

( ) ...|1

charge

charge discharge

charge

1 if ( | ) 0 if

pX S p

Nonparametric models» Kernel regression

• Weighted average of neighboring points

• Limited to low dimensional problems

» Locally linear methods• Dirichlet process mixtures• Radial basis functions

» Splines» Support vector machines» Deep neural networks

Parametric vs. nonparametric» Working definition of nonparametric:

• Given an infinitely large dataset, a nonparametric model can provide a perfect fit.

» “Parametric” models are inherently low dimensional.• Might have hundreds of parameters, perhaps more.• Includes “small” neural networks.

» “Nonparametric” models are inherently high dimensional.

• Deep neural networks may have tens of thousands of parameters. In principle can fit anything given a large enough dataset.

Parametric vs. sampled distributions» A sampled representation of a distribution is a form of

nonparametric model

Parametric distribution (pdf) Nonparametric distribution (cdf)

221( )

( )f x ( )F x

Parametric vs. nonparametric

» Robust CFAs are parametric» Scenario trees are nonparametric

True function

Nonparametric fit

Parametric fit

Observations

Price of product

Approximating strategiesNotes:» World of “big data” encourages use of high-dimensional

architectures:• Kernel regression• Locally linear models• Support vector machines• Deep neural networks

» Modern uses of neural networks (e.g. image recognition) depend on massive datasets (e.g. millions of records).

» In stochastic optimization, we are often estimating functions iteratively, starting with little or no data and growing.

» There are many applications where we are limited to dozens of iterations, although we will have settings where we may run an algorithm hundreds or even thousands of iterations.

What we are going to focus on:» Lookup tables

• Frequentist vs. Bayesian• Independent beliefs• Correlated beliefs• Hierarchical representations (time permitting)

» Parametric models• Linear in the parameters• Nonlinear in the parameters

» Sampled models• This can be thought of as a nonparametric distribution.

» Recursive estimation• Classical machine learning is batch• Our algorithms will always require adaptive learning (known

as “online machine learning” in the computer science community).

Learning with lookup tables

Bayesian vs. frequentistIndependent beliefsCorrelated beliefs

Learning with lookup tablesNotation:

» 𝑥 ∈ 𝑋 𝑥 , 𝑥 , … , 𝑥 assume 𝑀 is “not too large”» 𝜇 – truth (random variable)» 𝜇 – estimate of 𝜇 after n experiments have been run» 𝑊 – results of n+1st experiment at 𝑥 𝑥

Examples:» 𝑥 is a drug for diabetes, 𝑊 is the reduction in blood sugar» 𝑥 is a lineup for a basketball team, 𝑊 is the points scored

minus points against during the game.» 𝑥 is a price, 𝑊 is the revenue » 𝑥 is the bid for an add on google, 𝑊 is the number of clicks» 𝑥 is buy and sell prices for charging/discharging a battery using

electricity from the grid, 𝑊 is net profit for a day.» 𝑥 is a movie to display on Netflix, 𝑊 is whether or not a user

chooses the movie.

The information and decision process

Where:Belief about the value of ,

. Initial beliefDecision made using information from first

experiments.Initial decision based on information in .Observation from experiment,

Learning with lookup tablesThe information process:

» We make the decision using the results of experiment n, and then run experiment n+1, from which we obtain a noisy observation .

» We use the convention that any variable index by n contains the information from

1,..., nW W

Frequentist view» Estimates are based purely on experimental

observations.» Does not require any prior knowledge; similarly, it is

not able to use any prior knowledge that might be available.

Recursive formulas

n W^n mean variance mean variance1 74 74.00 74.002 66 70.00 32.00 70.00 32.003 70 70.00 16.00 70.00 16.004 49 64.75 120.92 64.75 120.925 76 67.00 116.00 67.00 116.006 79 69.00 116.80 69.00 116.807 47 65.86 166.48 65.86 166.488 52 64.13 166.70 64.13 166.709 53 62.89 159.61 62.89 159.6110 64 63.00 142.00 63.00 142.00

batch recursive

Bayesian view» The Bayesian view treats the true value of

as a random variable with some assumed distribution.

» We start with a prior distribution of belief about unknown parameters, even before we have collected any data.

» Useful when we have prior knowledge about a problem:

• We may know something about how a patient might respond to a drug because of experience with other patients.

• We have an idea how the market will respond to the price of a book because we see how it responded to other similar books.

Learning with lookup tablesBelief state

• 𝐵 , 𝜇 , 𝛽

» Sometimes use 𝑆 – The notation 𝑆 will eventually be used for more general states, but 𝐵 is always the belief state. Many authors use the term “belief state.” We use belief state and knowledge state interchangeably.

Conjugacy» Conjugacy means the posterior distribution is in the same family

as the prior distribution.» Normal prior plus normal observation produces normal posterior.

So, “normal-normal” is called a conjugate family.

Learning with lookup tablesConditional expectations» We often have to take expectations given what we know at time 𝑛.

There are several ways to write this. Consider the estimate 𝜇 of the value of a choice 𝑥 after we have made 𝑛 observations.

» We use the convention that by indexing the estimate by 𝑛 that it has seen the observations 𝑊 , 𝑊 , … , 𝑊 .

» The following are equivalent:

𝔼 𝜇 𝔼 𝜇 𝑊 , … , 𝑊 𝔼 𝜇 𝑆

» This applies to variances, which are a form of expectation. Recall that we can write 𝑉𝑎𝑟 𝑋 𝔼 𝑋 𝔼 𝑋 . Similarly we can write

𝑉𝑎𝑟 𝜇 𝑉𝑎𝑟 𝜇 𝑊 , … , 𝑊 𝑉𝑎𝑟 𝜇 𝑆

» So what is 𝑉𝑎𝑟 𝜇 ?

Bayesian updating – independent beliefs» Setup:

• Assume that our belief about μ is normally distributed with initial distribution (the “prior”):

• μ ∼ N μ , σ ,

• Now assume we choose to run experiment x x (chosen based on what we know at time 0) and observe W .

• Assume that W is a noisy observation of a function f x :

W f x ε

where we might assume ε ∼ N 0, σ ).» We can show that if the prior belief about μ is normally

distributed, and we then make a noisy observation that is normally distributed, then the posterior distribution of belief is also normally distributed.

Bayesian updating – independent beliefs» It is helpful to introduce the idea of the precision. This is just the

inverse of the variance.» The precision of our observation noise is

𝜎» Let 𝜎 , be our estimate of the variance of 𝜇 after 𝑛 experiments.

The precision of our belief about 𝜇 after 𝑛 experiments is

𝜎 ,

» The updating equations for 𝜇 and 𝛽 are given by

𝜇 .

𝛽 𝛽 𝛽 .

Notes:» In theory, this only applies if our belief about 𝜇 is normally

distributed, and observation errors are normally distributed.» In practice, we almost always use the normal-normal model,

because of the central limit theorem.» The reason is that averages are approximately normal. That is, an

estimate

𝜇1𝑁 𝑊

is a random variable that is approximately normally distributed.» This convergence is very fast! An average will be approximately

normal for samples as small as 5-10.» To illustrate, if we flip a coin 50 times and total the number of

heads (=1) and tails (=0), we will get a number that is normally distributed with mean 50𝑝, and a variance 50𝑝 1 𝑝 where 𝑝.5. If we do this 51 times, the total is still normally distributed.

IllustrationMeasurementVariance 133.3333Precision 0.0075

n W^n mean variance precisionPrior 50.00 900.00 0.0011111 51 50.87 116.13 0.0086112 54 52.33 62.07 0.0161113 70 57.94 42.35 0.0236114 62 58.92 32.14 0.0311115 50 57.19 25.90 0.0386116 79 60.73 21.69 0.0461117 56 60.07 18.65 0.0536118 77 62.15 16.36 0.0611119 68 62.79 14.57 0.06861110 76 64.09 13.14 0.076111

Bayesian updating

The predictive variance:

Bayesian updating

Derivation of posterior distributionfor normal-normal model

Optional – not covered in lecture

Derivation of posterior

Note:» We are going to derive the very simple Bayesian

updating formulas for the mean and the precision.» The formulas are simple, but the derivation is a bit

messy, and you may feel as if you have landed in the middle of an ORF 309 problem set.

» This is the only time that we will do a derivation like this in class.

» It will be repeated on one problem set.» It will not be on the midterm.

∝ 𝑒𝑥𝑝1

2 𝛽 𝑤 𝑢 𝛽 𝑢 𝑢

∝ 𝑒𝑥𝑝1

2 𝛽 𝑤 2𝛽 𝑤𝑢 𝛽 𝑢 𝛽 𝑢 2𝛽 𝑢𝑢 𝛽 𝑢

∝ 𝑒𝑥𝑝1

2 𝛽 𝛽 𝑢 2 𝛽 𝑤 𝛽 𝑢 𝑢 … 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

∝ 𝑒𝑥𝑝1

2 𝛽 𝛽 𝑢 2𝛽 𝑤 𝛽 𝑢

𝛽 𝛽 𝑢𝛽 𝑤 𝛽 𝑢

𝛽 𝛽

∝ 𝑒𝑥𝑝1

2 𝛽 𝛽 𝑢 𝑢

Where: 𝑢 . Now let 𝛽 𝛽 𝛽 .

Derivation of posteriorWhere: 𝑢 . Now let 𝛽 𝛽 𝛽 . This gives

∝ 𝑒𝑥𝑝1

2 𝛽 𝛽 𝑢 𝑢

∝ 𝑒𝑥𝑝1

2 𝛽 𝑢 𝑢

This looks just like a normal distribution:1

2𝜋𝜎𝑒𝑥𝑝

𝑥 𝜇𝜎

Now convert to precisions

𝛽2𝜋 𝑒𝑥𝑝

12 𝛽 𝑥 𝜇

This gives us our normalizing constant, so our posterior distribution is

𝑔 𝑢 𝑤𝛽2𝜋 𝑒𝑥𝑝

12 𝛽 𝑤 𝑢

… which is the density for a normal distribution.

Notes:» The “normal-normal” updating equations tend to be

used most of the time regardless of the underlying distributions of the random variables, for two reasons:

» 1) The distribution of an average is approximately normal due to the central limit theorem. In fact, the normal is a good approximation of an average when you have samples as small as 5 to 10 observations.

» 2) If you have a normal prior (see (1) above) and you then observe some random outcome (this could even be 0 or 1) is normal (see (1) above).

» One exception is when you are observing success/failure (0/1) outcomes, and the number of observations is small. In this case, we would use a beta Bernoulli distribution (we will do this later)

Lookup tables

Correlated beliefs

Lookup tables – correlated beliefs

Bayesian updating – correlated beliefs

Estimating continuous surfaces» Assume is a continuous vector, and

is a continuous function. Let be the random variable describing our belief about

» It is reasonable to assume that our belief about and is correlated as a function of the distance

.» Let be the covariance between our beliefs

about and . It is reasonable to assume that

Learning a two-dimensional surface

Notes:» Correlated beliefs is a powerful way of estimating

continuous surfaces with a relatively small number of observations.

» These updates require the use of a “distance function” that measures the distance between two points on the surface. Care needs to be used if you have more than about 5 dimensions.

» This logic can be used for categorical data:• Consumer choices based on age, location, gender, …• Performance of different materials that share characteristics.• Behavior of a drug as a function of patient characteristics.

Lookup tables

Derivation of posterior for normal-normal model. We will not cover this in lecture.

Do the algebra….» Replace variances with precisions.» Expand 2.48, obtaining six terms.» Collect terms involving – this will produce . Note

that is “u squared” while and for n=0, 1, …, represent estimates at time n.

» Collect terms involving “-2” – this will produce the term

» Remaining terms are not a function of , so you can just add whatever constant is needed to fill out the rest of the quadratic. The constant goes in the proportionality constant.

Bayesian updating for sampled models

Notes:» The “normal-normal” updating equations tend to be

used most of the time regardless of the underlying distributions of the random variables, for two reasons:

• 1) The distribution of an average is approximately normal due to the central limit theorem. In fact, the normal is a good approximation of an average when you have samples as small as 5 to 10 observations.

• 2) If you have a normal prior (see (1) above) and you then observe some random outcome (this could even be 0 or 1) is normal (see (1) above).

» One exception is when you are observing success/failure (0/1) outcomes, and the number of observations is small. In this case, we would use a beta-Bernoulli distribution (we will do this later).

Lookup tables

Hierarchical learning(I will just skim this in lecture)

Pre-decision state: we see the demands

Approximate dynamic programming

ˆ( , )t t

We use initial value function approximations…

0 ( ) 0V CO

0 ( ) 0V MN

$4500 ( ) 0V CA

0 ( ) 0V NY

ˆ( , )t t

… and make our first choice:

0 ( ) 0V CO

0 ( ) 0V CA

0 ( ) 0V NY

Approximate dynamic programming1x

0 ( ) 0V MN

Update the value of being in Texas.

1( ) 450V TX

0 ( ) 0V CO

0 ( ) 0V CA

0 ( ) 0V NY

0 ( ) 0V MN

Now move to the next state, sample new demands and make a new decision

0 ( ) 0V CO

0 ( ) 0V CA

0 ( ) 0V NY

1( ) 450V TX

1 1ˆ( , )

0 ( ) 0V MN

Update value of being in NY

0 ( ) 600V NY

0 ( ) 0V CO

0 ( ) 0V CA

1( ) 450V TX

1 ( )2

0 ( ) 0V MN

Move to California.

0 ( ) 0V CA

0 ( ) 0V CO

1( ) 450V TX

0 ( ) 600V NY

2 2ˆ( , )

0 ( ) 0V MN

Make decision to return to TX and update value of being in CA

0 ( ) 800V CA

0 ( ) 0V CO

1( ) 450V TX

0 ( ) 500V NY

2 2ˆ( , )

0 ( ) 0V MN

Updating the value function:

Old value: ( ) $450

New estimate:ˆ ( ) $800

How do we merge old with new?ˆ ( ) (1 ) ( ) ( ) ( )

(0.90)$450+(0.10)$800 $485

V TX V TX v TX

An updated value of being in TX

1( ) 485V TX

0 ( ) 0V CO 0 ( ) 600V NY

0 ( ) 800V CA

3 3ˆ( , )

0 ( ) 0V MN

Resource attribute:"State" that the trucker is currently ina

decision d

decision d’

'( )?dv a'

'( )?dv a

1( , )C a d 2( , )C a d'1( )V a

'2( )V a

NE regionPA

PA NEv v

Hierarchical aggregationDiscuss:

» Curse of dimensionality for lookup tables» Hierarchical aggregation:

• Ignoring attributes• Aggregating attributes

Examples:» Drugs and drug classes» Spatial» Temporal

Hierarchical aggregation

110 0 1 1 2 211 12

w V a w V w V aa

State dependent weight:

Computing bias and variance

Computing bias and variance:

The curse of dimensionality

The curse of dimensionalityThe curse of dimensionality» This is often confused with any vector-valued learning problem

(esp. in the context of dynamic programming).» It only arises when using lookup table representations of functions:

Dimensions

The curse of dimensionality“Solving” the curse of dimensionality is like inventing a perpetual motion machine

» The curse of dimensionality is a true curse. It cannot be “solved” without structure

• Convexity, linearity, monotonicity, …

» Claims to approximate functions without structure should be treated like perpetual motion machines.

The curse of dimensionality

Parametric models

Linear models

Linear modelsExamples:

» Independent variables, covariates, basis functions

Linear models

Linear modelsSparse models» Assume we approximate a function using a linear model

𝐹 𝑆 𝜃 𝜃 𝜙 𝑆∈

» Now consider what happens when the set of features 𝐹 is large. » Imagine we have a dataset consisting of 𝑓 , 𝑆 where 𝑓 is

the observed response given information in 𝑆 . We can fit our model using

min 𝑓 𝐹 𝑆 𝜃

» We may have too many features, but it is easily the case that we may have nonzero values of 𝜃 , even though there is no explanatory value.

Linear modelsSparse models» We can control how many elements of the vector 𝜃 are nonzero by

including a regularization factor:

min 𝑓 𝐹 𝑆 𝜃 𝜆 |𝜃 |

» We have to test different values of 𝜆, get the corresponding vector 𝜃 𝜆 , and then see how well this fits the data.

» Standard procedure is to fit on 80 percent of the data, test on 20 percent, and then rotate through the dataset five times so that each fifth of the data is used for testing at one point.

» Search over different values of 𝜆 to find the value of 𝜆 that produces the smallest error on the testing data.

» This is called cross validation.

Linear models

Recursive least squares-Stationary data

Parametric models

Nonlinear models

Nonlinear parametric models

Notes:» Maximum likelihood estimation for nonlinear models is

highly nonlinear and nonconvex. » These are often considered to be some of the hardest

classes of nonlinear programming problems.» We will introduce algorithms for this next week.

Sampled belief models

A sampled belief model

Concentration/temperature

0 ( )( | )1

0 1 2, ,i i i i

Thickness proportional to nkp

Possible experimental outcomes

Possibleresponses

Running an experiment – here we see the actual realization

Actual response

nx xThickness proportional to n

Updated posterior reflects proximity to each curve

n nx p

Notes:» Working with a sampled belief model can be

exceptionally easy and flexible.» We can apply rules to make sure we are creating valid

sampled estimates.• E.g. we might want to ensure that some coefficients are

positive.

» The challenge is that if is high-dimensional, a small sample (e.g. 100) may not contain any samples that are close to the true optimal solution.

» There are strategies that involve periodically re-generating new samples that can overcome this.

Neural networks

(only covered briefly in lecture)

Neural networksStart by illustrating with linear model

Neural networks

Notes:» Neural networks are very high-dimensional

architectures» When used with classical batch learning, they require

very large datasets.» There are settings where we can use iterative learning

(e.g. for value functions and policies), but these may require many millions of iterations (e.g. for deep neural networks).

» Shallow neural networks have been popular for many years for deterministic engineering control problems.

» Risk of overfitting stochastic problems is high.

Case study with neural networks

Neural network case study

Basic idea» Hotels have a range of room layouts as well as

reservation plans, all with their own rates. But one is known as the “BAR” rate (best available rate).

» We assume that the hotel has worked out a reasonable solution of how the different room rate plans should be priced relative to the BAR rate.

» The “practical policy” is to create a lookup table policy that maps the “state” (what we know) to an adjustment of the BAR rate.

• State variable requires using expert judgment to identify important factors

• State variables require combining numerical and qualitative factors (such as competition).

Zak’s rules» Developed by award-winning Revenue Manager Zak

Ali from YieldPlanet» Lookup table approach to price-setting» Derived from 6 binary factors and time to stay date

• Based on intuition and refined through observations• Not gradient-based: same effect regardless of small changes

within individual factors• Minimal look-back: Depends heavily on comparing data from

“this year” (TY) to “last year” (LY).

ABOBTY vs. ABOBLY» ABOB = Actual Business on the Books

• Refers to the number of reservations made for the stay date

» Compare to the equivalent date last year at the same number of days prior to arrival

APRICETY vs. APRICELY» APRICE = Actual Price» Takes into account the current BAR (Best Available

Rate), determined from business booked and forecasting relative to budget

Average Weekly Pickup TY vs. LY» The average rate that business is booked for stay date

• Use data from the previous 7 days

FTY vs. FLY» Forecast this year is total final occupancy forecasted in

terms of number of rooms booked for stay date• Based on (more quantitatively derived) forecasting estimates

calculated by dividing customers into segments• Done by separate forecasting team

» Forecast last year is final realized occupancy

COMPPRICE vs. MYPRICE» COMPPRICE = Prices of competition» How do we determine competitors?

• Difficult question to answer• Geography, Star Rating, Similar Price Range?• Historically been based on qualitative/anecdotal evidence

Is Price Elastic?» Elasticity: % Change in Quantity / % Change in Price» In practice, harder to determine

• Use price range of competition for comparable room type/rate plan combination

• Competition not always available (or could be irrelevant due to special convention or sold out status)

Neural network case studyPricing spreadsheet» First six columns determine “state”» Last column is recommended price above/below “BAR” rate

Example» Assume we are 5 days away from stay date

» Use numbers to compute state• If ABOBTY > ABOBLY then • If APRICETY > APRICELY then • Pickup TY < Pickup LY then • FTY > FLY then • COMPRICE > MYPRICE then • Unit demand is elastic [>1] then

Sample data

1 1, else = -1tS 2 1, else = -1tS

3 1, else = -1tS 4 1, else = -1tS

5 1, else = -1tS 6 1, else = -1tS

1 2 3 4 5 6( , , , , , ) "state" at time t t t t t t tS S S S S S S t

( ) 1 ( )BAR ZAKt tP S p p S

The lookup table policy for Zak’s rules:

Zak’s rules:» Use binary factors and number of days prior to arrival

(0-3, 4-7, 8-14, etc.) to determine percentage increase/decrease in price

» Price changes become more dramatic closer to stay date• -20% to 20% range in lookup table corresponding to 0-3 days

prior to arrival• -8% to 8% range on decision tree corresponding to 71-90 days

prior to arrival

» The numbers themselves are more based on intuition rather than quantitative analysis

• Percentages are typically integer increases/decreases, rather than more precise numerical estimates

Neural network for pricing hotel rooms

Notes:» This neural network, which is quite small, was fitted on

a dataset of state-action pairs constructed with the expert “Zak.”

» The graphs that follow show the behavior of the neural network as a function of different inputs (e.g. price, hotel type, ..)

» The curves are not well behaved. What is happening here is that there is overfitting. Even though this is a small neural network, the number of parameters is still large relative to the size of the dataset.

Nonparametric models

Nonparametric methods

Definition:» A nonparametric model has the property that as the

amount of data goes to infinity, modeling errors go to zero.

Illustrate:» K-nearest neighbor» Kernel regression

Kernels

Parametric vs. nonparametric

» Robust CFAs are parametric» Scenario trees are nonparametric

True function

Nonparametric fit

Parametric fit

Observations

Price of product

Nonparametric methodsLookup table belief models (Frazier)» Independent beliefs

» Correlated beliefsLinear, parametric belief models (Frazier)

Nonparametric models» Hierarchical aggregation (Mes)» Kernel regression (Barut)

Local parametric models» Dirichlet clouds with RBF (Jamshidi)» KG derivation (Harvey Cheng)

Generalized linear models» KG derivation (Si Chen)

1 2 3 4 5

Dirichlet process mixtures of generalized linear regression models (Hannah, Blei and WBP, 2011).» Very general, but very slow.» Cannot be implemented recursively.

Dirichlet clouds using radial basis functions (Jamshidi and WBP, 2012)

» Local parametric approximations.» Produces variable order approximation that

adapts to the function.

Discussion:» Local smoothing methods struggle with curse of

dimensionality. In high dimensions, no two data points are ever “close.”

» Nonparametric representations are very high-dimensional, which makes it hard to “store” a model.

More advanced methods» Deep neural networks

• Before, we described neural networks as a nonlinear parametric model.

• Deep neural networks, which have the property of being able to approximate any function, are classified as nonparametric.

• These have proven to be very powerful on image processing and voice recognition, but not in stochastic optimization.

» Support vector machines (classification)» Support vector regression (continuous)

• We have had surprising but limited success with SVM, but considerably more empirical research is needed.

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